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Diffstat (limited to 'source/know/concept/cauchy-principal-value')
| -rw-r--r-- | source/know/concept/cauchy-principal-value/index.md | 18 |
1 files changed, 9 insertions, 9 deletions
diff --git a/source/know/concept/cauchy-principal-value/index.md b/source/know/concept/cauchy-principal-value/index.md index a2582f2..f09611b 100644 --- a/source/know/concept/cauchy-principal-value/index.md +++ b/source/know/concept/cauchy-principal-value/index.md @@ -7,15 +7,15 @@ categories: layout: "concept" --- -The **Cauchy principal value** $\mathcal{P}$, +The **Cauchy principal value** $$\mathcal{P}$$, or just **principal value**, is a method for integrating problematic functions, i.e. functions with singularities, whose integrals would otherwise diverge. -Consider a function $f(x)$ with a singularity at some finite $x = b$, +Consider a function $$f(x)$$ with a singularity at some finite $$x = b$$, which is hampering attempts at integrating it. -To resolve this, we define the Cauchy principal value $\mathcal{P}$ as follows: +To resolve this, we define the Cauchy principal value $$\mathcal{P}$$ as follows: $$\begin{aligned} \boxed{ @@ -24,8 +24,8 @@ $$\begin{aligned} } \end{aligned}$$ -If $f(x)$ instead has a singularity at postive infinity $+\infty$, -then we define $\mathcal{P}$ as follows: +If $$f(x)$$ instead has a singularity at postive infinity $$+\infty$$, +then we define $$\mathcal{P}$$ as follows: $$\begin{aligned} \boxed{ @@ -34,10 +34,10 @@ $$\begin{aligned} } \end{aligned}$$ -And analogously for $-\infty$. -If $f(x)$ has singularities both at $+\infty$ and at $b$, +And analogously for $$-\infty$$. +If $$f(x)$$ has singularities both at $$+\infty$$ and at $$b$$, then we simply combine the two previous cases, -such that $\mathcal{P}$ is given by: +such that $$\mathcal{P}$$ is given by: $$\begin{aligned} \mathcal{P} \int_{a}^\infty f(x) \:dx @@ -49,5 +49,5 @@ And so on, until all problematic singularities have been dealt with. In some situations, for example involving the [Sokhotski-Plemelj theorem](/know/concept/sokhotski-plemelj-theorem/), -the symbol $\mathcal{P}$ is written without an integral, +the symbol $$\mathcal{P}$$ is written without an integral, in which case the calculations are implicitly integrated. |